One studies the codimension two unfolding of an orbit homoclinic to a saddle-node equilibrium in . The eigenvalues of the equilibrium are supposed to be positive and negative real parts, except for a simple eigenvalue . Several new methods are employed. The bifurcation diagram is obtained by the bifurcation function which is derived by the method of Lyapunov-Schmidt decomposition. The idea of exponential trichotomy is employed to study the linearized equation around
The method of cross sections and Poincaré mappings are used to study the bifurcation of the periodic orbits from
Under the Poincaré mapping, submanifolds of certain type, called the u- slices can be found to be fixed and hence bifurcation of periodic orbits is proved.